TY - JOUR
T1 - The Convexity of Inclusions and Gradient’s Concentration for Lamé Systems with Partially Infinite Coefficients
AU - Hou , Yuanyuan
AU - Li , Haigang
JO - Analysis in Theory and Applications
VL - 3
SP - 240
EP - 310
PY - 2024
DA - 2024/10
SN - 40
DO - http://doi.org/10.4208/ata.OA-2023-0019
UR - https://global-sci.org/intro/article_detail/ata/23467.html
KW - Gradient estimates, convex inclusions, blowup analysis, linear elasticity, composite materials.
AB - <p style="text-align: justify;">It is interesting to study the stress concentration between two adjacent stiff
inclusions in composite materials, which can be modeled by the Lamé system with partially infinite coefficients. To overcome the difficulty from the lack of maximum
principle for elliptic systems, we use the energy method and an iteration technique to
study the gradient estimates of the solution. We first find a novel phenomenon that
the gradient will not blow up any more once these two adjacent inclusions fail to be
locally relatively strictly convex, namely, the top and bottom boundaries of the narrow
region are partially “flat”. In order to further explore the blow-up mechanism of the
gradient, we next investigate two adjacent inclusions with relative convexity of order $m$ and finally reveal an underlying relationship between the blow-up rate of the stress
and the order of the relative convexity of the subdomains in all dimensions.</p>